Mathematics Abstracts of Presentations

Four Solutions of a Geometry Problem about Angles on Lattices

Toh Pee Choon

Helen is a primary school student who was stuck on a geometry problem. She asked Ivan, a mathematics expert from her coin-collectors club, to help her with the problem. Ivan proposed four different approaches to the problem, ranging from elementary to advanced, using techniques in trigonometry, vectors and even number theory. The different approaches will be discussed in this talk.

Arc reversals of cycles in orientations of G vertex-multiplications

Willie Wong Han Wah

One of the early problems concerning arc reversals is: Given two tournaments of the same order, is it possible to obtain one from the other by a sequence of a prescribed type of arc reversals? Let $C_k$ denote a directed cycle of length $k$. In 1964, Ryser gave an affirmative answer of using $C_3$ for any two tournaments with the same score sequence. Later, Waldrop gave an independent proof and further established two results in which $“C_3”$ is replaced by $“C_4”$ and by $“C_5”$. Beineke and Moon proved the $C_4$-equivalence for any two bipartite tournaments with the same score lists. In this talk, we discuss some extensions of these results to multipartite tournaments and more broadly, orientations of $G$ vertex-multiplications.

Sparse fused group lasso models for spectroscopic data

Soh Chin Gi

High-dimensional spectroscopic data has applications in many fields such as food science, forensic science and biomedical science due to the information it provides about the chemical compositions of the samples. The fitting of classification and regression models to such data is a challenging task due to the high-dimensional setting, as well as the issue of high correlation between spectral variables. This talk presents a regularized model for spectroscopic data that incorporates sparsity, smoothness and group structure. The results from some simulation studies on the use of the regularized model will be discussed. An application of the model to Fourier-transform infrared spectroscopic data adulteration studies in olive oil will also be presented. The results suggest that the sparse fused group lasso is able to achieve good prediction performance, while improving on the interpretability of the resulting models.

Similarities in geometry

Zhao Dongsheng

The notion of similar triangles plays a very important role in geometry. There are several different conditions for similarity. In this talk, we shall
(1) examine the relationships among such conditions,
(2) show some applications of similar triangles both in mathematics theory and the solution of practical problems, and
(3) explore the possibility of extending this notion to other classes of objects.